20.5. The Convolution Theorem. Introduction. Prerequisites. Learning Outcomes
|
|
- Marian Copeland
- 5 years ago
- Views:
Transcription
1 The Convolution Theorem 2.5 Introduction In this Section we introduce the convolution of two functions f(t), g(t) which we denote by (f g)(t). The convolution is an important construct because of the convolution theorem which allows us to find the inverse Laplace transform of a product of two transformed functions: L 1 {F (s)g(s)} (f g)(t) Prerequisites Before starting this Section you should... Learning Outcomes On completion you should be able to... HELM (28): Section 2.5: The Convolution Theorem be able to find Laplace transforms and inverse Laplace transforms of simple functions be able to integrate by parts understand how to use step functions in integration calculate the convolution of simple functions apply the convolution theorem to obtain inverse Laplace transforms 55
2 1. Convolution Let f(t) and g(t) be two functions of t. The convolution of f(t) and g(t) is also a function of t, denoted by (f g)(t) and is defined by the relation (f g)(t) f(t x)g(x) dx However if f and g are both causal functions then (strictly) f(t), g(t) are written f(t)u(t) and g(t)u(t) respectively, so that (f g)(t) f(t x)u(t x)g(x)u(x) dx f(t x)g(x) dx because of the properties of the step functions: u(t x) if x > t and u(x) if x <. Key Point 14 Convolution If f(t) and g(t) are causal functions then their convolution is defined by: (f g)(t) f(t x)g(x) dx This is an odd looking definition but it turns out to have considerable use both in Laplace transform theory and in the modelling of linear engineering systems. The reader should note that the variable of integration is x. As far as the integration process is concerned the t-variable is (temporarily) regarded as a constant. Example 3 Find the convolution of f and g if f(t) tu(t) and g(t) t 2 u(t). Solution Therefore f(t x) (t x)u(t x) and g(x) x 2 u(x) (f g)(t) (t x)x 2 dx 1 3 x3 t 1 4 x4 ] t 1 3 t4 1 4 t t4 56 HELM (28): Workbook 2: Laplace Transforms
3 Example 4 Find the convolution of f(t) t.u(t) and g(t) sin t.u(t). Solution Here f(t x) (t x)u(t x) and g(x) sin x.u(x) and so (f g)(t) (t x) sin x dx We need to integrate by parts. We find, remembering again that t is a constant in the integration process, so that (t x) sin x dx ] t (t x) cos x + t] t sin x ] t cos x dx t sin t ( 1)( cos x) dx (f g)(t) t sin t or, equivalently, in this case (t sin t)(t) t sin t Task In Example 4 we found the convolution of f(t) t.u(t) and g(t) sin t.u(t). In this Task you are asked to find the convolution (g f)(t) that is, to reverse the order of f and g. Begin by writing (g f)(t) as an appropriate integral: g(t x) sin(t x).u(t x) and f(x) xu(x), so (g f)(t) sin(t x).x dx HELM (28): Section 2.5: The Convolution Theorem 57
4 Now evaluate the convolution integral: (g f)(t) sin(t x).x dx ] t x cos(t x) ] t t ] + sin(t x) cos(t x) dx t sin t This Task illustrates the general result in the following Key Point: Key Point 15 Commutativity Property of Convolution (f g)(t) (g f)(t) In words: the convolution of f(t) with g(t) is the same as the convolution of g(t) with f(t). Task Obtain the Laplace transforms of f(t) t.u(t) and g(t) sin t.u(t) and (f g)(t). Begin by finding L{f(t)}, L{g(t)}: L{f(t)} 1 s 2 L{g(t)} 1 s (from Table 1) 58 HELM (28): Workbook 2: Laplace Transforms
5 Now find L{(f g)(t)}: From Example 4 (f g)(t) t sin t and so L{(f g)(t)} L{t sin t} 1 s 2 1 s Now compare L{f(t)} L{g(t)} with L{f g(t)}. What do you observe? L{(f g)(t)} 1 s 1 2 s ( ) 1 L{f(t)}L{g(t)} F (s)g(s) s 2 s We see that the Laplace transform of the convolution of f(t) and g(t) is the product of their separate Laplace transforms. This, in fact, is a general result which is expressed in the statement of the convolution theorem which we discuss in the next subsection. 2. The convolution theorem Let f(t) and g(t) be causal functions with Laplace transforms F (s) and G(s) respectively, i.e. L{f(t)} F (s) and L{g(t)} G(s). Then it can be shown that Key Point 16 The Convolution Theorem L 1 {F (s)g(s)} (f g)(t) or equivalently L{(f g)(t)} F (s)g(s) HELM (28): Section 2.5: The Convolution Theorem 59
6 Example 5 Find the inverse transform of (a) Using partial fractions 6 s(s 2 + 9). (b) Using the convolution theorem. Solution 6 (a) s(s 2 + 9) (2/3) (2/3)s and so s s { } { } { } L s 2 s(s ) L s L 1 2 s u(t) 2 cos 3t.u(t) 3 (b) Let us choose So F (s) 2 s and G(s) 3 s then f(t) L 1 {F (s)} 2u(t) and g(t) L 1 {G(s)} sin 3t.u(t) L 1 {F (s)g(s)} (f g)(t) (by the convolution theorem) 2u(t x) sin 3x.u(x) dx Now the variable t can take any value from to +. If t < then the variable of integration, x, is negative and so u(x). We conclude that (f g)(t) if t < that is, (f g)(t) is a causal function. Let us now consider the other possibility for t, that is the range t. Now, in the range of integration x t and so u(t x) 1 u(x) 1 since both t x and x are non-negative. Therefore L 1 {F (s)g(s)} 2 sin 3x dx 2 ] t 3 cos 3x 2 (cos 3t 1) t 3 Hence L 1 6 { s(s 2 + 9) } 2 (cos 3t 1)u(t) 3 which agrees with the value obtained above using the partial fraction approach. 6 HELM (28): Workbook 2: Laplace Transforms
7 Task Use the convolution theorem to find the inverse transform of s H(s) (s 1)(s 2 + 1). Begin by choosing two functions of s, that is, F (s) and G(s): Although there are many possibilities it would seem sensible to choose F (s) 1 s 1 and G(s) s s since, by inspection, we can write down their inverse Laplace transforms: f(t) L 1 {F (s)} e t u(t) and g(t) L 1 {G(s)} cos t.u(t) Now construct the convolution integral: h(t) h(t) L 1 {H(s)} L 1 {F (s)g(s)} f(t x)g(x) dx e t x u(t x) cos x.u(x) dx HELM (28): Section 2.5: The Convolution Theorem 61
8 Now complete the evaluation of the integral, treating the cases t < and t separately: You should find h(t) 1 2 (sin t cos t + et )u(t) since h(t) if t < and h(t) e t x cos x dx if t ] t e t x sin x ( 1)e t x sin x dx sin t + e t x cos x ] t sin t cos t + e t h(t) or 2h(t) sin t cos t + e t t Finally h(t) 1 2 (sin t cos t + et )u(t) ( e t x )( cos x) dx (integrating by parts) 1. Find the convolution of Exercises (a) 2tu(t) and t 3 u(t) (b) e t u(t) and tu(t) (c) e 2t u(t) and e t u(t). In each case reverse the order to check that (f g)(t) (g f)(t). 2. Use the convolution theorem to determine the inverse Laplace transforms of (a) s 1 s 2 (s + 1) (b) 1 (s 1)(s 2) (c) 1 (s 2 + 1) 2 1. (a) 1 1 t5 (b) t 1 + e t (c) e t e 2t 2. (a) (t 1 + e t )u(t) (b) ( e t + e 2t )u(t) (c) 1 (sin t t cos t)u(t) 2 62 HELM (28): Workbook 2: Laplace Transforms
20.3. Further Laplace Transforms. Introduction. Prerequisites. Learning Outcomes
Further Laplace Transforms 2.3 Introduction In this Section we introduce the second shift theorem which simplifies the determination of Laplace and inverse Laplace transforms in some complicated cases.
More information20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes
Transfer Functions 2.6 Introduction In this Section we introduce the concept of a transfer function and then use this to obtain a Laplace transform model of a linear engineering system. (A linear engineering
More informationChapter DEs with Discontinuous Force Functions
Chapter 6 6.4 DEs with Discontinuous Force Functions Discontinuous Force Functions Using Laplace Transform, as in 6.2, we solve nonhomogeneous linear second order DEs with constant coefficients. The only
More information11.5. The Chain Rule. Introduction. Prerequisites. Learning Outcomes
The Chain Rule 11.5 Introduction In this Section we will see how to obtain the derivative of a composite function (often referred to as a function of a function ). To do this we use the chain rule. This
More information23.4. Convergence. Introduction. Prerequisites. Learning Outcomes
Convergence 3.4 Introduction In this Section we examine, briefly, the convergence characteristics of a Fourier series. We have seen that a Fourier series can be found for functions which are not necessarily
More information11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes
Parametric Differentiation 11.6 Introduction Sometimes the equation of a curve is not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this Section we see how to calculate the
More informationComputing inverse Laplace Transforms.
Review Exam 3. Sections 4.-4.5 in Lecture Notes. 60 minutes. 7 problems. 70 grade attempts. (0 attempts per problem. No partial grading. (Exceptions allowed, ask you TA. Integration table included. Complete
More information11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes
Parametric Differentiation 11.6 Introduction Often, the equation of a curve may not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this section we see how to calculate the
More information16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this Section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
More informationDifferential Equations
Differential Equations Math 341 Fall 21 MWF 2:3-3:25pm Fowler 37 c 21 Ron Buckmire http://faculty.oxy.edu/ron/math/341/1/ Worksheet 29: Wednesday December 1 TITLE Laplace Transforms and Introduction to
More informationOrdinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International
More informationMATH 251 Examination II April 4, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 4, 2016 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationMA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.)
MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.) Lecture 19 Lecture 19 MA 201, PDE (2018) 1 / 24 Application of Laplace transform in solving ODEs ODEs with constant coefficients
More information2.4. Characterising Functions. Introduction. Prerequisites. Learning Outcomes
Characterising Functions 2.4 Introduction There are a number of different terms used to describe the ways in which functions behave. In this Section we explain some of these terms and illustrate their
More informationMATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit
More information= e t sin 2t. s 2 2s + 5 (s 1) Solution: Using the derivative of LT formula we have
Math 090 Midterm Exam Spring 07 S o l u t i o n s. Results of this problem will be used in other problems. Therefore do all calculations carefully and double check them. Find the inverse Laplace transform
More information7.3. Determinants. Introduction. Prerequisites. Learning Outcomes
Determinants 7.3 Introduction Among other uses, determinants allow us to determine whether a system of linear equations has a unique solution or not. The evaluation of a determinant is a key skill in engineering
More informationMATH 251 Examination II April 3, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 3, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More information(f g)(t) = Example 4.5.1: Find f g the convolution of the functions f(t) = e t and g(t) = sin(t). Solution: The definition of convolution is,
.5. Convolutions and Solutions Solutions of initial value problems for linear nonhomogeneous differential equations can be decomposed in a nice way. The part of the solution coming from the initial data
More information+ + LAPLACE TRANSFORM. Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions.
COLOR LAYER red LAPLACE TRANSFORM Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions. + Differentiation of Transforms. F (s) e st f(t)
More informationMATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
More information22.2. Applications of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes
Applications of Eigenvalues and Eigenvectors 22.2 Introduction Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration
More informationReview Sol. of More Long Answer Questions
Review Sol. of More Long Answer Questions 1. Solve the integro-differential equation t y (t) e t v y(v)dv = t; y()=. (1) Solution. The key is to recognize the convolution: t e t v y(v) dv = e t y. () Now
More informationLecture 29. Convolution Integrals and Their Applications
Math 245 - Mathematics of Physics and Engineering I Lecture 29. Convolution Integrals and Their Applications March 3, 212 Konstantin Zuev (USC) Math 245, Lecture 29 March 3, 212 1 / 13 Agenda Convolution
More informationExponential and Logarithmic Functions
Contents 6 Exponential and Logarithmic Functions 6.1 The Exponential Function 2 6.2 The Hyperbolic Functions 11 6.3 Logarithms 19 6.4 The Logarithmic Function 27 6.5 Modelling Exercises 38 6.6 Log-linear
More informationLaplace Theory Examples
Laplace Theory Examples Harmonic oscillator s-differentiation Rule First shifting rule Trigonometric formulas Exponentials Hyperbolic functions s-differentiation Rule First Shifting Rule I and II Damped
More informationLaplace Transform. Chapter 4
Chapter 4 Laplace Transform It s time to stop guessing solutions and find a systematic way of finding solutions to non homogeneous linear ODEs. We define the Laplace transform of a function f in the following
More informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 6. Solve the completely inhomogeneous diffusion problem on the half-line
Strauss PDEs 2e: Section 3.3 - Exercise 2 Page of 6 Exercise 2 Solve the completely inhomogeneous diffusion problem on the half-line v t kv xx = f(x, t) for < x
More informationMATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section:
MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all
More informationLater in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.
10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing
More informationSolutions to Assignment 7
MTHE 237 Fall 215 Solutions to Assignment 7 Problem 1 Show that the Laplace transform of cos(αt) satisfies L{cosαt = s s 2 +α 2 L(cos αt) e st cos(αt)dt A s α e st sin(αt)dt e stsin(αt) α { e stsin(αt)
More informationMath 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington.
Math 307 Lecture 19 Laplace Transforms of Discontinuous Functions W.R. Casper Department of Mathematics University of Washington November 26, 2014 Today! Last time: Step Functions This time: Laplace Transforms
More informationThe Laplace Transform and the IVP (Sect. 6.2).
The Laplace Transform and the IVP (Sect..2). Solving differential equations using L ]. Homogeneous IVP. First, second, higher order equations. Non-homogeneous IVP. Recall: Partial fraction decompositions.
More informationMATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must
More informationMath 201 Lecture 17: Discontinuous and Periodic Functions
Math 2 Lecture 7: Dicontinuou and Periodic Function Feb. 5, 22 Many example here are taken from the textbook. he firt number in () refer to the problem number in the UA Cutom edition, the econd number
More informationOrdinary differential equations
Class 11 We will address the following topics Convolution of functions Consider the following question: Suppose that u(t) has Laplace transform U(s), v(t) has Laplace transform V(s), what is the inverse
More informationMATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:
MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationThe Laplace transform
The Laplace transform Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Laplace transform Differential equations 1
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More information30.4. Matrix Norms. Introduction. Prerequisites. Learning Outcomes
Matrix Norms 304 Introduction A matrix norm is a number defined in terms of the entries of the matrix The norm is a useful quantity which can give important information about a matrix Prerequisites Before
More informationExam 2 Study Guide: MATH 2080: Summer I 2016
Exam Study Guide: MATH 080: Summer I 016 Dr. Peterson June 7 016 First Order Problems Solve the following IVP s by inspection (i.e. guessing). Sketch a careful graph of each solution. (a) u u; u(0) 0.
More informationMA 201, Mathematics III, July-November 2016, Laplace Transform
MA 21, Mathematics III, July-November 216, Laplace Transform Lecture 18 Lecture 18 MA 21, PDE (216) 1 / 21 Laplace Transform Let F : [, ) R. If F(t) satisfies the following conditions: F(t) is piecewise
More informationLimit Laws- Theorem 2.2.1
Limit Laws- Theorem 2.2.1 If L, M, c, and k are real numbers and lim x c f (x) =L and lim g(x) =M, then x c 1 Sum Rule: lim x c (f (x)+g(x)) = L + M 2 Difference Rule: lim x c (f (x) g(x)) = L M 3 Constant
More informationMathematical Methods and its Applications (Solution of assignment-12) Solution 1 From the definition of Fourier transforms, we have.
For 2 weeks course only Mathematical Methods and its Applications (Solution of assignment-2 Solution From the definition of Fourier transforms, we have F e at2 e at2 e it dt e at2 +(it/a dt ( setting (
More information7.2. Matrix Multiplication. Introduction. Prerequisites. Learning Outcomes
Matrix Multiplication 7.2 Introduction When we wish to multiply matrices together we have to ensure that the operation is possible - and this is not always so. Also, unlike number arithmetic and algebra,
More informationf(t)e st dt. (4.1) Note that the integral defining the Laplace transform converges for s s 0 provided f(t) Ke s 0t for some constant K.
4 Laplace transforms 4. Definition and basic properties The Laplace transform is a useful tool for solving differential equations, in particular initial value problems. It also provides an example of integral
More informationIntroduction. f(t) dt. lim. f(t) dt = lim. 1 (1+t)1 p. dt; here p>0. h = (1+h)1 p 1 p. dt = (1+t) p 1 p
Introduction Method of Laplace transform is very useful in solving differential equations. Its main feature is to convert a differential equation into an algebric equation, which allows us to obtain the
More informationFall 2001, AM33 Solution to hw7
Fall 21, AM33 Solution to hw7 1. Section 3.4, problem 41 We are solving the ODE t 2 y +3ty +1.25y = Byproblem38x =logt turns this DE into a constant coefficient DE. x =logt t = e x dt dx = ex = t By the
More information2.4. Characterising functions. Introduction. Prerequisites. Learning Outcomes
Characterising functions 2.4 Introduction There are a number of different terms used to describe the ways in which functions behave. In this section we explain some of these terms and illustrate their
More informationFind the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.
Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no
More informationMATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:
MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be
More informationStudy Guide Block 2: Ordinary Differential Equations. Unit 9: The Laplace Transform, Part Overview
Unit 9: The Laplace Transform, Part 1 1. Overview ~ The Laplace transform has application far beyond its present role in this block of being a useful device for solving certain types of linear differential
More information10.1 Complex Arithmetic Argand Diagrams and the Polar Form The Exponential Form of a Complex Number De Moivre s Theorem 29
10 Contents Complex Numbers 10.1 Complex Arithmetic 2 10.2 Argand Diagrams and the Polar Form 12 10.3 The Exponential Form of a Complex Number 20 10.4 De Moivre s Theorem 29 Learning outcomes In this Workbook
More informationDepartment of Mathematics. MA 108 Ordinary Differential Equations
Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai 476, INDIA. MA 8 Ordinary Differential Equations Autumn 23 Instructor Santanu Dey Name : Roll No : Syllabus and Course Outline
More informationBabu Banarasi Das Northern India Institute of Technology, Lucknow
Babu Banarasi Das Northern India Institute of Technology, Lucknow B.Tech Second Semester Academic Session: 2012-13 Question Bank Maths II (EAS -203) Unit I: Ordinary Differential Equation 1-If dy/dx +
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More information6.2. The Hyperbolic Functions. Introduction. Prerequisites. Learning Outcomes
The Hyperbolic Functions 6. Introduction The hyperbolic functions cosh x, sinh x, tanh x etc are certain combinations of the exponential functions e x and e x. The notation implies a close relationship
More information17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes
Parametric Curves 7.3 Introduction In this Section we eamine et another wa of defining curves - the parametric description. We shall see that this is, in some was, far more useful than either the Cartesian
More informationAPPM 2360: Midterm exam 3 April 19, 2017
APPM 36: Midterm exam 3 April 19, 17 On the front of your Bluebook write: (1) your name, () your instructor s name, (3) your lecture section number and (4) a grading table. Text books, class notes, cell
More informationMidterm Exam, Thursday, October 27
MATH 18.152 - MIDTERM EXAM 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Midterm Exam, Thursday, October 27 Answer questions I - V below. Each question is worth 20 points, for a total of
More information30.5. Iterative Methods for Systems of Equations. Introduction. Prerequisites. Learning Outcomes
Iterative Methods for Systems of Equations 0.5 Introduction There are occasions when direct methods (like Gaussian elimination or the use of an LU decomposition) are not the best way to solve a system
More informationAnswer Key b c d e. 14. b c d e. 15. a b c e. 16. a b c e. 17. a b c d. 18. a b c e. 19. a b d e. 20. a b c e. 21. a c d e. 22.
Math 20580 Answer Key 1 Your Name: Final Exam May 8, 2007 Instructor s name: Record your answers to the multiple choice problems by placing an through one letter for each problem on this answer sheet.
More informationf(x)e ikx dx. (19.1) ˆf(k)e ikx dk. (19.2)
9 Fourier transform 9 A first look at the Fourier transform In Math 66 you all studied the Laplace transform, which was used to turn an ordinary differential equation into an algebraic one There are a
More informationapplications of integration 2
Contents 15 applications of integration 1. Integration of vectors. Calculating centres of mass 3. Moment of inertia Learning outcomes In this workbook you will learn to interpret an integral as the limit
More informationName: Solutions Exam 3
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More informationCoordinate goemetry in the (x, y) plane
Coordinate goemetr in the (x, ) plane In this chapter ou will learn how to solve problems involving parametric equations.. You can define the coordinates of a point on a curve using parametric equations.
More informationMathQuest: Differential Equations
MathQuest: Differential Equations Laplace Tranforms 1. True or False The Laplace transform method is the only way to solve some types of differential equations. (a) True, and I am very confident (b) True,
More informationMaximum and Minimum Values section 4.1
Maximum and Minimum Values section 4.1 Definition. Consider a function f on its domain D. (i) We say that f has absolute maximum at a point x 0 D if for all x D we have f(x) f(x 0 ). (ii) We say that f
More informationMATH 251 Examination II April 7, 2014 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 7, 2014 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationPractice Exercises on Differential Equations
Practice Exercises on Differential Equations What follows are some exerices to help with your studying for the part of the final exam on differential equations. In this regard, keep in mind that the exercises
More information9 More on the 1D Heat Equation
9 More on the D Heat Equation 9. Heat equation on the line with sources: Duhamel s principle Theorem: Consider the Cauchy problem = D 2 u + F (x, t), on x t x 2 u(x, ) = f(x) for x < () where f
More informationMa 221 Final Exam Solutions 5/14/13
Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes
More informationA New Numerical Scheme for Solving Systems of Integro-Differential Equations
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 1, No. 2, 213, pp. 18-119 A New Numerical Scheme for Solving Systems of Integro-Differential Equations Esmail Hesameddini
More informationApplied Differential Equation. October 22, 2012
Applied Differential Equation October 22, 22 Contents 3 Second Order Linear Equations 2 3. Second Order linear homogeneous equations with constant coefficients.......... 4 3.2 Solutions of Linear Homogeneous
More informationSection Vector Functions and Space Curves
Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of
More informationThird In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix
Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.
More informationMath 308 Exam II Practice Problems
Math 38 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationECE : Linear Circuit Analysis II
Purdue University School of Electrical and Computer Engineering ECE 20200 : Linear Circuit Analysis II Summer 2014 Instructor: Aung Kyi San Instructions: Midterm Examination I July 2, 2014 1. Wait for
More informationEE2090 Basic Engineering Mathematics. Assoc. Prof. M. Adams Room : S2.2-B2-23 Extn :
EE2090 Basic Engineering Mathematics Assoc. Prof. M. Adams Room : S2.2-B2-23 Extn : 4361 Email : eadams@ntu.edu.sg TOPICS 1. Multiple Integrals 2. Infinite Sequences & Series 3. Vectors 4. Introduction
More informationMath 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question
More information11.4. Differentiating ProductsandQuotients. Introduction. Prerequisites. Learning Outcomes
Differentiating ProductsandQuotients 11.4 Introduction We have seen, in the first three Sections, how standard functions like n, e a, sin a, cos a, ln a may be differentiated. In this Section we see how
More information21.4. Engineering Applications of z-transforms. Introduction. Prerequisites. Learning Outcomes
Engineering Applications of z-transforms 21.4 Introduction In this Section we shall apply the basic theory of z-transforms to help us to obtain the response or output sequence for a discrete system. This
More informationName: Solutions Exam 4
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More informationFundamental Solution
Fundamental Solution onsider the following generic equation: Lu(X) = f(x). (1) Here X = (r, t) is the space-time coordinate (if either space or time coordinate is absent, then X t, or X r, respectively);
More informationMath 334 Fall 2011 Homework 10 Solutions
Nov. 5, Math 334 Fall Homework Solution Baic Problem. Expre the following function uing the unit tep function. And ketch their graph. < t < a g(t = < t < t > t t < b g(t = t Solution. a We
More informationISE I Brief Lecture Notes
ISE I Brief Lecture Notes 1 Partial Differentiation 1.1 Definitions Let f(x, y) be a function of two variables. The partial derivative f/ x is the function obtained by differentiating f with respect to
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this section we analse curves in the local neighbourhood of a stationar point and, from this analsis, deduce necessar conditions satisfied b local maima and local minima.
More informationwe get y 2 5y = x + e x + C: From the initial condition y(0) = 1, we get 1 5 = 0+1+C; so that C = 5. Completing the square to solve y 2 5y = x + e x 5
Math 24 Final Exam Solution 17 December 1999 1. Find the general solution to the differential equation ty 0 +2y = sin t. Solution: Rewriting the equation in the form (for t 6= 0),we find that y 0 + 2 t
More informationAbout the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project
About the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project undertaken by a consortium of five English universities led by
More informationI Laplace transform. I Transfer function. I Conversion between systems in time-, frequency-domain, and transfer
EE C128 / ME C134 Feedback Control Systems Lecture Chapter 2 Modeling in the Frequency Domain Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley Lecture
More information23.6. The Complex Form. Introduction. Prerequisites. Learning Outcomes
he Complex Form 3.6 Introduction In this Section we show how a Fourier series can be expressed more concisely if we introduce the complex number i where i =. By utilising the Euler relation: e iθ cos θ
More informationWave Equation With Homogeneous Boundary Conditions
Wave Equation With Homogeneous Boundary Conditions MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 018 Objectives In this lesson we will learn: how to solve the
More informationNumerical Methods of Approximation
Contents 31 Numerical Methods of Approximation 31.1 Polynomial Approximations 2 31.2 Numerical Integration 28 31.3 Numerical Differentiation 58 31.4 Nonlinear Equations 67 Learning outcomes In this Workbook
More informationMath 266 Midterm Exam 2
Math 266 Midterm Exam 2 March 2st 26 Name: Ground Rules. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use one 4-by-6
More information14.4. Lengths of Curves and Surfaces of Revolution. Introduction. Prerequisites. Learning Outcomes
Lengths of Curves and Surfaces of Revolution 4.4 Introduction Integration can be used to find the length of a curve and the area of the surface generated when a curve is rotated around an axis. In this
More informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
More informationName: ID.NO: Fall 97. PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS.
MATH 303-2/6/97 FINAL EXAM - Alternate WILKERSON SECTION Fall 97 Name: ID.NO: PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS. Problem
More informationMath 216 Final Exam 14 December, 2017
Math 216 Final Exam 14 December, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More information